a study of 50 people living in crescent beach, bc, showed the average age as 42 years with a standard…

Explanation:

Step1: Find critical value

For a 95% confidence interval, the critical value (z) (assuming normal distribution) is (z = 1.96) (from standard normal distribution tables).

Step2: Calculate margin of error for (n = 50)

The formula for margin of error (E=z\times\frac{\sigma}{\sqrt{n}}). Given (\sigma = 12), (n = 50). [ E_1=1.96\times\frac{12}{\sqrt{50}}\approx1.96\times\frac{12}{7.07}\approx1.96\times1.697\approx3.33 ] The confidence interval is (\bar{x}\pm E_1), where (\bar{x} = 42). So the interval is ((42 - 3.33,42 + 3.33)=(38.67,45.33))

Step3: Calculate margin of error for (n = 100)

Using the same formula (E=z\times\frac{\sigma}{\sqrt{n}}), with (n = 100) [ E_2=1.96\times\frac{12}{\sqrt{100}}=1.96\times1.2 = 2.35 ] The confidence interval is (\bar{x}\pm E_2). So the interval is ((42- 2.35,42 + 2.35)=(39.65,44.35))

Answer:

a) The 95% confidence interval is ((38.67,45.33)) b) When (n) increases from (50) to (100), the confidence interval narrows. This is because as the sample size (n) increases, the margin of error (E = z\times\frac{\sigma}{\sqrt{n}}) decreases (since (\sqrt{n}) increases). A larger sample size provides more information and reduces the uncertainty, resulting in a more precise estimate of the population mean.